Abstract

We have developed a powerful 3D Monte Carlo code, as part of the Radiance in a Dynamic Ocean (RaDyO) project, which can compute the complete effective Mueller matrix at any detector position in a completely inhomogeneous turbid medium, in particular, a coupled atmosphere-ocean system. The light source can be either passive or active. If the light source is a beam of light, the effective Mueller matrix can be viewed as the complete impulse response Green matrix for the turbid medium. The impulse response Green matrix gives us an insightful way to see how each region of a turbid medium affects every other region. The present code is validated with the multicomponent approach for a plane-parallel system and the spherical harmonic discrete ordinate method for the 3D scalar radiative transfer system. Furthermore, the impulse response relation for a box-type cloud model is studied. This 3D Monte Carlo code will be used to generate impulse response Green matrices for the atmosphere and ocean, which act as inputs to a hybrid matrix operator–Monte Carlo method. The hybrid matrix operator–Monte Carlo method will be presented in part II of this paper.

© 2008 Optical Society of America

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References

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  1. A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
    [CrossRef]
  2. J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transf. 58, 379-398 (1997).
    [CrossRef]
  3. K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
    [CrossRef]
  4. Y. Chen, K. N. Liou, and Y. Gu, “An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 92, 189-200 (2005).
    [CrossRef]
  5. L. G. Stenholm, H. Störzer, and R. Wehrse, “An efficient method for the solution of 3-D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transf. 45, 47-56 (1991).
    [CrossRef]
  6. R. Cahalan, W. Ridgway, and W. Wiscombe, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776-3790 (1994).
    [CrossRef]
  7. D. M. O'Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41-59 (1992).
    [CrossRef]
  8. L. Roberti and C. Kummerow, “Monte Carlo calculations of polarized microwave radiation emerging from cloud structures,” J. Geophy. Res. 104, 2093-2104 (1999).
    [CrossRef]
  9. A. Battaglia and S. Mantovani, “Forward Monte Carlo computations of fully polarized microwave radiation in non-isotropic media,” J. Quant. Spectrosc. Radiat. Transf. 95, 285-308(2005).
    [CrossRef]
  10. Y. Chen and K. N. Liou, “A Monte Carlo method for 3D thermal infrared radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 101, 166-178 (2006).
    [CrossRef]
  11. R. F. Cahalan, L. Oreopoulos, A. Marshak, K. F. Evans, A. B. Davis, R. Pincus, K. H. Yetzer, B. Mayer, R. Davies, T. P. Ackerman, H. W. Barker, E. E. Clothiaux, R. G. Ellingson, M. J. Garay, E. Kassianov, S. Kinne, A. Macke, W. O'Hirok, P. T. Partain, S. M. Prigarin, A. N. Rublev, G. L. Stephens, F. Szczap, E. E. Takara, T. Vrnai, G. Wen, and T. B. Zhuravleva, “The 13RC: bringing together the most advanced radiative transfer tools for cloudy atmospheres,” Bull. Am. Meteorol. Soc. 86, 1275-1293 (2005).
    [CrossRef]
  12. G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).
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    [CrossRef]
  14. H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400-412 (2001).
    [CrossRef]
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  17. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
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  19. D. G. Collins, W. G. Blattner, M. B. Wells, and H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of the radiation emerging from spherical-shell atmospheres,” Appl. Opt. 11, 2684-2696 (1972).
    [CrossRef] [PubMed]
  20. G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453-1472 (1989).
    [CrossRef]
  21. H. H. Tynes, “Monte Carlo solutions of the radiative transfer equation for scattering systems,” Ph.D. dissertation (Texas A&M University, 2001), p. 46.
  22. Concise Dictionary of Scientific Biography (Scribner, 1981), p. 643. Willebrord Snel von Royen used only one l in his last name.
  23. R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, 1965).
  24. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  25. E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803-2812 (1993).
    [CrossRef] [PubMed]
  26. E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19-31 (1996).
    [CrossRef]
  27. Q. Liu, C. Simmer, and E. Ruprcht, “Three-dimensional radiative transfer effects of clouds in the microwave spectral range,” J. Geophys. Res. 101, 4289-4298 (1996).
    [CrossRef]

2006

Y. Chen and K. N. Liou, “A Monte Carlo method for 3D thermal infrared radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 101, 166-178 (2006).
[CrossRef]

2005

R. F. Cahalan, L. Oreopoulos, A. Marshak, K. F. Evans, A. B. Davis, R. Pincus, K. H. Yetzer, B. Mayer, R. Davies, T. P. Ackerman, H. W. Barker, E. E. Clothiaux, R. G. Ellingson, M. J. Garay, E. Kassianov, S. Kinne, A. Macke, W. O'Hirok, P. T. Partain, S. M. Prigarin, A. N. Rublev, G. L. Stephens, F. Szczap, E. E. Takara, T. Vrnai, G. Wen, and T. B. Zhuravleva, “The 13RC: bringing together the most advanced radiative transfer tools for cloudy atmospheres,” Bull. Am. Meteorol. Soc. 86, 1275-1293 (2005).
[CrossRef]

A. Battaglia and S. Mantovani, “Forward Monte Carlo computations of fully polarized microwave radiation in non-isotropic media,” J. Quant. Spectrosc. Radiat. Transf. 95, 285-308(2005).
[CrossRef]

Y. Chen, K. N. Liou, and Y. Gu, “An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 92, 189-200 (2005).
[CrossRef]

2001

1999

L. Roberti and C. Kummerow, “Monte Carlo calculations of polarized microwave radiation emerging from cloud structures,” J. Geophy. Res. 104, 2093-2104 (1999).
[CrossRef]

1998

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

1997

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transf. 58, 379-398 (1997).
[CrossRef]

L. Roberti, “Monte Carlo radiative transfer in the microwave and in the visible: biasing techniques,” Appl. Opt. 36, 7929-7938 (1997).
[CrossRef]

1996

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19-31 (1996).
[CrossRef]

Q. Liu, C. Simmer, and E. Ruprcht, “Three-dimensional radiative transfer effects of clouds in the microwave spectral range,” J. Geophys. Res. 101, 4289-4298 (1996).
[CrossRef]

1994

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

R. Cahalan, W. Ridgway, and W. Wiscombe, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776-3790 (1994).
[CrossRef]

1993

1992

D. M. O'Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41-59 (1992).
[CrossRef]

1991

L. G. Stenholm, H. Störzer, and R. Wehrse, “An efficient method for the solution of 3-D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transf. 45, 47-56 (1991).
[CrossRef]

1989

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453-1472 (1989).
[CrossRef]

1972

1954

C. Cox and W. Munk, “Statistics of sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

1852

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

J. Quant. Spectrosc. Radiat. Transf.

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transf. 58, 379-398 (1997).
[CrossRef]

Appl. Opt.

Atmos. Res.

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

Bull. Am. Meteorol. Soc.

R. F. Cahalan, L. Oreopoulos, A. Marshak, K. F. Evans, A. B. Davis, R. Pincus, K. H. Yetzer, B. Mayer, R. Davies, T. P. Ackerman, H. W. Barker, E. E. Clothiaux, R. G. Ellingson, M. J. Garay, E. Kassianov, S. Kinne, A. Macke, W. O'Hirok, P. T. Partain, S. M. Prigarin, A. N. Rublev, G. L. Stephens, F. Szczap, E. E. Takara, T. Vrnai, G. Wen, and T. B. Zhuravleva, “The 13RC: bringing together the most advanced radiative transfer tools for cloudy atmospheres,” Bull. Am. Meteorol. Soc. 86, 1275-1293 (2005).
[CrossRef]

J. Atmos. Sci.

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

R. Cahalan, W. Ridgway, and W. Wiscombe, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776-3790 (1994).
[CrossRef]

J. Geophy. Res.

L. Roberti and C. Kummerow, “Monte Carlo calculations of polarized microwave radiation emerging from cloud structures,” J. Geophy. Res. 104, 2093-2104 (1999).
[CrossRef]

J. Geophys. Res.

Q. Liu, C. Simmer, and E. Ruprcht, “Three-dimensional radiative transfer effects of clouds in the microwave spectral range,” J. Geophys. Res. 101, 4289-4298 (1996).
[CrossRef]

J. Mar. Res.

C. Cox and W. Munk, “Statistics of sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

J. Quant. Spectrosc. Radiat. Transf.

A. Battaglia and S. Mantovani, “Forward Monte Carlo computations of fully polarized microwave radiation in non-isotropic media,” J. Quant. Spectrosc. Radiat. Transf. 95, 285-308(2005).
[CrossRef]

Y. Chen and K. N. Liou, “A Monte Carlo method for 3D thermal infrared radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 101, 166-178 (2006).
[CrossRef]

D. M. O'Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 48, 41-59 (1992).
[CrossRef]

Y. Chen, K. N. Liou, and Y. Gu, “An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transf. 92, 189-200 (2005).
[CrossRef]

L. G. Stenholm, H. Störzer, and R. Wehrse, “An efficient method for the solution of 3-D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transf. 45, 47-56 (1991).
[CrossRef]

E. P. Zege and L. I. Chaikovskaya, “New approach to the polarized radiative transfer problem,” J. Quant. Spectrosc. Radiat. Transf. 55, 19-31 (1996).
[CrossRef]

Limnol. Oceanogr.

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453-1472 (1989).
[CrossRef]

Trans. Cambridge Philos. Soc.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Other

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

H. H. Tynes, “Monte Carlo solutions of the radiative transfer equation for scattering systems,” Ph.D. dissertation (Texas A&M University, 2001), p. 46.

Concise Dictionary of Scientific Biography (Scribner, 1981), p. 643. Willebrord Snel von Royen used only one l in his last name.

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, 1965).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

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Figures (13)

Fig. 1
Fig. 1

Example of the model of the medium used in the 3D Monte Carlo code. Inhomogeneous layers are divided into voxels with different optical properties; however, each voxel is optically homogeneous.

Fig. 2
Fig. 2

Estimate due to transmission of the interface.

Fig. 3
Fig. 3

Radiance at four locations as a function of the zenith angle for the three-layer plane-parallel system with total optical depth of 1.25. The optical depths of the Rayleight atmosphere and ocean are 0.25 and 1.0, respectively. Both the Rayleigh atmosphere and ocean are conservative. The albedo of the Lambertian bottom is 1.0 also. A solar light source is used.

Fig. 4
Fig. 4

Second element of the radiance vector, Q, divided by the radiance I at the four locations as a function of the zenith angle for the system as shown in Fig. 3.

Fig. 5
Fig. 5

(a) Reflected radiance distribution at the top of the Gaussian medium from the 3D Monte Carlo code. The observer direction is θ = 0 ° , ϕ = 0 ° . The solar source is at an angle of θ = 135 ° , ϕ = 0 ° . The base grid is 21 × 21 × 10 and the total number of photon histories is 2.0 × 10 7 . (b) The absolute difference in radiance between the 3D Monte Carlo code and SHDOM for this case. The number of discrete ordinates for SHDOM is N μ = 16 , N ϕ = 32 .

Fig. 6
Fig. 6

Schematic of the pp case in the x z plane. A detector is placed at the top of the scattering medium with a coordinate ( 0 , 0 , 10 m ) . The scattering medium is a two layer system. The top layer is for a HG phase function of thickness 5 m and the lower layer for a Rayleigh phase function of thickness 5 m . The positions of the incident impulses are at x = 0 , 1 , 2 , 3 , 4 , and 5 m .

Fig. 7
Fig. 7

Impulse response relation for a detector shown in Fig. 6.

Fig. 8
Fig. 8

Same as Fig. 7 except β HG = 0.98 m - 1 .

Fig. 9
Fig. 9

Schematic representation of the 3D case. A detector is placed at the top of the scattering medium with a coordinate ( 0 , 0 , 10 m ) . The scattering medium is a single HG ( g = 0.86 ) cloud with dimension 1 m × 1 m × 5 m surrounded by a Rayleigh medium. β HG = 0.18 m - 1 , β Rayleigh = 0.02 m - 1 . The positions of the incident impulses are at x = 0 , 1 , 2 , 3 , 4 , and 5 m . The geometric size of the HG cloud in the figure is not proportional to its actual size.

Fig. 10
Fig. 10

Impulse response relation for a detector shown in Fig. 9. The label of the curves shows the position of the incident impulse.

Fig. 11
Fig. 11

Same as Fig. 10 except β HG = 0.98 m - 1 .

Fig. 12
Fig. 12

Radiances measured by a detector at the top of the scattering medium with a coordinate ( 0 , 0 , 10 m ) . β HG = 0.18 m - 1 , β Rayleigh = 0.02 m - 1 . See the text for the label of the cases.

Fig. 13
Fig. 13

Same as Fig. 12 except β HG = 0.98 m - 1 .

Equations (25)

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τ = - ln ( ζ ) ,
τ = - ln ( 1 - [ 1 - exp ( - τ 0 ) ] ζ ) ,
w = w [ 1 - exp ( - τ 0 ) ] ,
d = i d i , τ = i τ i = i β i d i ,
r = r 0 + d · n ,
w = w ω 0 .
M j ( r ) = w exp ( - τ d ) | cos ( θ d ) | R ( π - i 2 ) P ( Θ s , r ) R ( - i 1 ) M p ,
( 1 0 0 0 0 cos 2 ψ sin 2 ψ 0 0 - sin 2 ψ cos 2 ψ 0 0 0 0 1 ) ,
z = tan η = σ ( - 2 ln ζ 1 ) , ϕ = 2 π ζ 2 ,
M j ( r ) = n 2 exp ( - τ 2 d ) · R ( π - i 2 ) T f ( θ i , θ t ) R ( - i 1 ) · w exp ( - τ 12 ) | cos ( θ 12 ) | R ( π - i 2 ) P ( Θ , r 1 ) R ( - i 1 ) M p ,
ζ 1 = 1 μ p ( μ , r 1 ) d μ , μ = cos ( θ s ) , φ s = 2 π ζ 2 ,
M p = R ( π - i 2 ) P ˜ ( θ s , r 1 ) R ( - i 1 ) M p = R ( π - i 2 ) P ( θ s , r 1 ) R ( - i 1 ) M p / p ( θ s , r 1 ) .
w = w ω b .
P ( θ d ) = cos ( θ d ) π ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) .
cos ( θ s ) = ζ 1 , φ s = 2 π ζ 2 ,
R f = ( α + η α - η 0 0 α - η α + η 0 0 0 0 γ re - γ im 0 0 γ im γ re ) ,
T f = ( α + η α - η 0 0 α - η α + η 0 0 0 0 γ re - γ im 0 0 γ im γ re ) ,
α = 1 2 [ tan ( θ i - θ t ) tan ( θ i + θ t ) ] 2 , η = 1 2 [ sin ( θ i - θ t ) sin ( θ i + θ t ) ] 2 , α = 1 2 [ 2 sin θ t cos θ i sin ( θ i + θ t ) cos ( θ i - θ t ) ] 2 , η = 1 2 [ 2 sin θ t cos θ i sin ( θ i + θ t ) ] 2 , γ re = - tan ( θ i - θ t ) sin ( θ i - θ t ) tan ( θ i + θ t ) sin ( θ i + θ t ) , γ re = 4 sin 2 ( θ t ) cos 2 ( θ i ) sin 2 ( θ i + θ t ) cos ( θ i - θ t ) ,
α = η = 1 2 ,
γ = n 2 cos ( θ i ) + i sin 2 θ i - n 2 n 2 cos ( θ i ) - i sin 2 θ i - n 2 cos ( θ i ) - i sin 2 θ i - n 2 cos ( θ i ) + i sin 2 θ i - n 2 ,
M eff ( r ) = 1 N P · A ( r ) j ( M j ( r ) + M j ( r ) ) ,
M eff = r M eff ( r ) · A ( r ) .
I s ( θ , ϕ , r ) = M eff ( θ , ϕ , r , θ 0 , ϕ 0 , r 0 ) · I i ( θ 0 , ϕ 0 , r 0 ) ,
I s ( θ , ϕ ) = M eff ( θ , ϕ , θ 0 , ϕ 0 ) · I i ( θ 0 , ϕ 0 ) .
β e = 4.63 exp ( - 4 ( x - 1 ) 2 - 4 ( y - 1 ) 2 - 16 ( z - 1 / 2 ) 2 ) m - 1

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